# Download Applied Linear Statistical Models 5th ed by M. Kutner, et al., PDF

By M. Kutner, et al.,

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Example text

Of having the same cardinality, the same number of elements. That notion, expressed in the language of bijections, is crucial in the classical cardinality theory as weIl as in the theory of generalized cardinals of fuzzy sets in Chapter 4. Its position in the presented theory of scalar cardinalities is much weaker. The scalar cardinality of a fuzzy set principally plays the role of an index of cardinality of that fuzzy set. An equipotency relation - such that for each A, BE FFS with a cardinality pattern 1 is generally difficult to defme, especially in terms of some bijections.

However, if 1 is finite and, say, 1 = {I, 2, ... ,k} with k ~ 1, let us introduce the following notation: ~ Ai := Al U fil Ai := Al jE! and jE! S A 2U S ••• U s Ak n t A 2 n t ... n t A k, where sand I are quite arbitrary. 9), we then get Aus ( ( ~ A)V jE! = n B) = n AusBi' jE! rul Ajv iE! with u := SV, jE! Ant ( ( fil U B) jE! A;)V = lj jE! (ViEl: AjcB) => ( fil iE! Aj c fil B j & jE! = U AntBi, iE! Aiv with v:= IV, JEJ ~ jE! Ai C ~ B j), iE! 4. Other Elements 0/ the Language 0/ Fuzzy Sets 'Vjel: (i) Ai e Aj e iEJ where V 31 eJ Ai' iEJ is a strong negation.

On the other hand, there exists another group of constructive approaches in which that cardinality is itself fuzzy. e. by means of some weighted family of usual cardinals with the weights from [0,1]. This alternative "fuzzy" optics leads to a very complete and adequate cardinal description of A at the price of relatively high complexity. Generalized cardinals will be denoted by lowercase letters a, ß, y, ... l from the first half of the Greek alphabet. If a generalized cardinal a does express the cardinality of A, we write 1 A 1 = a, and we say that the cardinality 01 A is equal to a.